= Solution
For a family in <stellar homology>, dimensionless radial profiles are fixed. <Mass conservation> and <hydrostatic equilibrium> then give the central scalings
$$
\rho_c\propto\frac{M}{R^3},
\qquad
\frac{P_c}{R}\sim\frac{GM\rho_c}{R^2},
\qquad
\boxed{P_c\propto\frac{GM\rho_c}{R}\propto\frac{GM^2}{R^4}}.
$$
The <ideal gas> equation of state, with fixed <mean molecular weight>, consequently gives
$$
\boxed{T_c\propto\frac{P_c}{\rho_c}\propto\frac{M}{R}},
$$
where fixed dimensional constants such as $G$ are suppressed in homology relations.
The <proton–proton chain> law $\epsilon\propto\rho T^{7/2}$ gives the nuclear luminosity
$$
L_{\rm nuc}\sim M\epsilon_c
\propto M\rho_cT_c^{7/2}
\propto M^{11/2}R^{-13/2}.
$$
On the other hand, <radiative diffusion in a star> gives
$$
L_{\rm rad}\propto\frac{RT_c^4}{\kappa_c\rho_c}.
$$
Using the <Kramers opacity law> $\kappa_c\propto\rho_cT_c^{-7/2}$,
$$
L_{\rm rad}\propto\frac{RT_c^{15/2}}{\rho_c^2}
\propto M^{11/2}R^{-1/2}.
$$
Thermal equilibrium requires $L_{\rm nuc}=L_{\rm rad}$. Their common mass factor cancels, leaving $R^{-13/2}\propto R^{-1/2}$, and hence
$$
\boxed{R=\text{constant}},
\qquad
\boxed{L\propto M^{11/2}}.
$$
This model captures the gas-pressure support, pp-chain burning, and strongly mass-dependent luminosity of the <lower main sequence>, including the Sun approximately. Its fully radiative assumption is an idealization: the Sun has a convective envelope, and sufficiently low-mass <red dwarfs> become largely or fully convective, so real radii are not exactly constant.
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